download the script: Definition of Entropy

Since the concept “reversibility” plays a crucial role in the thermodynamics, it is necessary for us to find an approach to determine whether the process reversible or not. And the most well-known Clausius Inequality provides us this possibility.

As we all know, in thermodynamics, if the cyclic integral of a parameter x is zero, that means:then this parameter x is a property of state and path-independent as well.

The Clausius Inequality indicates that for a reversible cycle process, we obtain an equation expressed as: As mentioned above, since the cyclic integral of quantity (δq_rev/T) is zero, this term also represents a property of state.

Clausius discovered therefore a new thermodynamic property of state which is called “Entropy” and written as S: (This definition is the macroscopic definition of entropy. For the engineer, we usually use this definition instead of microscopic definition)

• Entropy S is an extensive property of a system and has the unit of [S]=J/K

• Entropy per unit mass s is an intensive property of a system and has the unit of [S]=J/kg·K

Since entropy is the property of state and is a point function as well, we can determine the entropy change of a system which undergoes a process (reversible or irreversible) from state 1 to state 2 as:

We notice that the entropy change can be calculated only if the system undergoes a reversible process. But luckily, the entropy is defined as a property of state and the entropy change only depends on the initial state and the final state, therefore, even though a system undergoes an irreversible process from state 1 to state 2, we can arbitrarily draw any other convenient imaginary reversible paths from 1 to 2 to calculate the entropy change.

But for an irreversible process, ds≠δq/T. According to the Clausius Inequality for an irreversible process, we have: Now assume that the system goes along a cycle process which consists of one irreversible process 1→2 (path L) and one reversible process 2→1 (path M). Therefore:

here:

• δq_rev means heat transferred during an irreversible process (path M)

• δq_irr means heat transferred during a reversible process (path L)

Since 2→1 is reversible, the second term on the left side of the above inequality is:

Hence we now obtain:

We can now draw a conclusion: for an irreversible process, the entropy change of a closed system from state 1 to state 2 is greater than the integral of (δq_irr/T)

The difference of Δs and (δq_irr/T) ascribes to the entropy generation. (The entropy generation is discussed here: https://thermodynamics-engineer.com/the-increase-of-entropy-change/)